Definición de metamathematics en inglés:

plural sustantivo

Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics.

Proof theory, a form of metamathematics, studies the ways in which proofs are used in mathematics.

Thus, only in his early sixties did Hilbert truly proceed to create proof theory and metamathematics.